Comparing the Riskiness of Dependent Portfolios

Abstract

A nonparametric test based on nested L-statistics and designed to compare the riskiness of portfolios was introduced by Brazauskas, Jones, Puri, and Zitikis (2007). Its asymptotic and small-sample properties were primarily explored for independent portfolios, though independence is not a required condition for the test to work. In this dissertation, we investigate how the performance of the test changes when insurance portfolios are dependent. To achieve that goal, we perform a simulation study where we consider three different risk measures: conditional tail expectation, proportional hazards transform, and mean. Further, three portfolios are generated from exponential, Pareto, and lognormal distributions, and their interdependence is modeled with the three-dimensional t and Gaussian copulas. It is found that the presence of comonotonicity makes the test very liberal for all the risk measures under consideration. For various other types of dependence, the results are mixed, i.e., they depend on the chosen risk measure, sample size, and even on the test’s significance level. We illustrate how to incorporate such findings into sensitivity analysis of the decisions. The risks we analyze represent tornado damages in different regions of the United States from 1890 to 1999. In addition, we provide a theoretical explanation to the behavior of the power function of the test by considering the usual stochastic orders of the Gini indexes of multivariate normal risks with the same marginals but different dependence structures. Finally, we generalize the comparison for the Gini indexes of multivariate elliptical risks.